1. StatementsReasons Given: ZY || WX; WX  ZY Prove:  W   Y WX YZWARM UP- Complete the Proof. 1. WX  ZY, ZY || WX 2.  YZX   WXZ 3. ZX  ZX 4. ΔYZX  ΔWXZ 5.  W   Y 1. Given 3. Reflexive Property 2. If lines are parallel, alternate interior angles are congruent. 4. SAS Postulate 5. CPCTC

3. Definition – A quadrilateral with both pairs of opposite sides parallel.

4. Theorem 5-1: Opposite sides of a parallelogram are congruent. EF G H EF  GH, FG  EH

5. PROOF OF THEOREM 5-1:E F G H 1 2 3 4 Given: EFGH Prove: EF  GH, FG  EH 2. EF || GH, FG || HE2. Def. of parallelogram 5. ΔEFH  ΔGHF5. ASA Postulate 6. CPCTC6. EF  GH; FG  HE 3.  1   4,  2   3 4. FH  FH4. Reflexive Property 3. If lines are parallel, then alternate interior angles are congruent. 1. EFGH1. Given

6. Theorem 5-2: Opposite angles of a parallelogram are congruent. EF G H  F   H,  E   G

7. PROOF OF THEOREM 5-2: E F G H 4 1 2 3 Given: EFGH Prove:  E   G 2. EF || HG, EH || FG2. Def. of parallelogram 5. ΔEFH  ΔGHF5. ASA Postulate 6. CPCTC6.  E   G 3.  4   3,  2   1 4. FH  FH4. Reflexive Property 3. If lines are parallel, then alternate interior angles are congruent. 1. EFGH1. Given

8. E F G H 2. EF || HG2. Def. of parallelogram 3.  E and  H are supplementary 4. m  E + m  H = 180 3. If lines are parallel, same-side interior angles are supplementary. 1.  E   G, EFGH1. Given m  F + m  G = 180 5. m  E + m  H = m  F + m  G5. Substitution 4. Definition of Supp. Angles  F and  G are supplementary 6. m  H = m  F,  H   F6. Subtraction How can we prove the other opposite angles are congruent using information from the previous proof? Given: EFGH,  E   G Prove:  H   F

9. Consecutive angles of a parallelogram are supplementary. EF G H  E and  H are supplementary  F and  G are supplementary  E and  F are supplementary  H and  G are supplementary

10. Theorem 5-3: Diagonals of a parallelogram bisect each other. QR S T M QM  MS, RM  MT

11. PROOF OF THEOREM 5-3: Given: QRST Prove: QS and RT bisect each other 2. QR || ST2. Def. of parallelogram 5. ΔQRM  ΔSTM5. ASA Postulate 6. CPCTC6. QM  SM; RM  TM 3.  1   2,  3   4 4. QR  ST 4. Opposite sides of a parallelogram are congruent. 3. If lines are parallel, then alternate interior angles are congruent. 1. QRST1. Given QR S T M 1 2 3 4 8. Def. of Segment Bisector 8. QS and RT bisect each other 7. Definition of a Midpoint 7. M is the midpoint of QS and RT

12. Name all the properties of a parallelogram. 2 pairs of opposite sides are parallel 2 pairs of opposite sides are congruent 2 pairs of opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other

13. CLASS WORK Complete 5-1 Class Work worksheet and turn it in when you are finished.